Morphisms generated by functions Given a function $f: A \to B$, I can construct a morphism $g : A^* \to B^*$ where $X^*$ denotes some free structure generated by $X$ (Could be monoid, group, module, etc.).  
I'd like to study morphisms generated by functions a bit more, but I'm not sure where I should be looking.  Do they have a specific name?  Where would I look in an algebra/category theory book for more info?
 A: Let $\mathcal C$ be some category of algebraic structures.
Define the forgetful functor $U : \mathcal C \to \mathbf {Set}$.
Construct its left adjoint $F \dashv U$.
Then $F$ takes sets to freely generated objects and morphisms between the generators to morphisms between freely generated objects.
A: You should look in category theory books.
There, you would read something along these lines:
Given a set A you can build a group (or a monoid, module, ect) which is called the free group and is denoted by $F(A)$ (or even $FA$, which I find very bad, personally).
Given a function $f:A \to B$ you can build a group-morphism (or monoid, module,.., -morphism) $F(f):F(A) \to F(B)$ between the free groups generated by $A$ and $B$. 
The $F$ operator is called the Free functor from $\mathcal{Set}$ - the category of sets and functions - to  $\mathcal{Grp}$ -the category of groups and group-morphisms.
$$F:\mathcal{Set} \to \mathcal{Grp}$$
The interesting thing about this operator is that it can be proved that it is a functor.
Correspondence with your notation:
$F(X) \longleftrightarrow X^*$
$F(f) \longleftrightarrow g$
As was mentioned by @user58512, it turns out that $F$ is the left adjoint to the $U$ functor. 
